Andrew Pitts, from the Journal of Information and Computation, available from Andrew Pitts’s website:

This paper formalises within first-order logic some common practices in computer science to do with representing and reasoning about syntactical structures involving lexically scoped binding constructs. It introduces *Nominal Logic*, a version of first-order many-sorted logic with equality containing primitives for renaming via name-swapping, for freshness of names, and for name-binding. Its axioms express properties of these constructs satisfied by the *FM-sets* model of syntax involving binding, which was recently introduced by the author and M.J. Gabbay and makes use of the Fraenkel-Mostowski permutation model of set theory. Nominal Logic serves as a vehicle for making two general points. First, name-swapping has much nicer logical properties than more general, non-bijective forms of renaming while at the same time providing a sufficient foundation for a theory of structural induction/recursion for syntax modulo α-equivalence. Secondly, it is useful for the practice of operational semantics to make explicit the *equivariance property* of assertions about syntax – namely that their validity is invariant under name-swapping.

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Strange, I thought nominal logic used “∇”, not “ᴎ”? Regardless, here’s how to define a function counting variable occurrences using the work described in the paper.